.TH  CTFTRI 1 "November 2008" " LAPACK routine (version 3.2)                                    " " LAPACK routine (version 3.2)                                    " 
.SH NAME
CTFTRI - computes the inverse of a triangular matrix A stored in RFP format
.SH SYNOPSIS
.TP 19
SUBROUTINE CTFTRI(
TRANSR, UPLO, DIAG, N, A, INFO )
.TP 19
.ti +4
CHARACTER
TRANSR, UPLO, DIAG
.TP 19
.ti +4
INTEGER
INFO, N
.TP 19
.ti +4
COMPLEX
A( 0: * )
.SH PURPOSE
CTFTRI computes the inverse of a triangular matrix A stored in RFP
format.
This is a Level 3 BLAS version of the algorithm.
.br
.SH ARGUMENTS
.TP 10
TRANSR    (input) CHARACTER
= \(aqN\(aq:  The Normal TRANSR of RFP A is stored;
.br
= \(aqC\(aq:  The Conjugate-transpose TRANSR of RFP A is stored.
.TP 8
UPLO    (input) CHARACTER
.br
= \(aqU\(aq:  A is upper triangular;
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= \(aqL\(aq:  A is lower triangular.
.TP 8
DIAG    (input) CHARACTER
.br
= \(aqN\(aq:  A is non-unit triangular;
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= \(aqU\(aq:  A is unit triangular.
.TP 8
N       (input) INTEGER
The order of the matrix A.  N >= 0.
.TP 8
A       (input/output) COMPLEX array, dimension ( N*(N+1)/2 );
On entry, the triangular matrix A in RFP format. RFP format
is described by TRANSR, UPLO, and N as follows: If TRANSR =
.br
\(aqN\(aq then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
.br
(0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = \(aqC\(aq then RFP is
the Conjugate-transpose of RFP A as defined when
TRANSR = \(aqN\(aq. The contents of RFP A are defined by UPLO as
follows: If UPLO = \(aqU\(aq the RFP A contains the nt elements of
upper packed A; If UPLO = \(aqL\(aq the RFP A contains the nt
elements of lower packed A. The LDA of RFP A is (N+1)/2 when
TRANSR = \(aqC\(aq. When TRANSR is \(aqN\(aq the LDA is N+1 when N is
even and N is odd. See the Note below for more details.
On exit, the (triangular) inverse of the original matrix, in
the same storage format.
.TP 8
INFO    (output) INTEGER
= 0: successful exit
.br
< 0: if INFO = -i, the i-th argument had an illegal value
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> 0: if INFO = i, A(i,i) is exactly zero.  The triangular
matrix is singular and its inverse can not be computed.
.SH FURTHER DETAILS
We first consider Standard Packed Format when N is even.
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We give an example where N = 6.
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    AP is Upper             AP is Lower
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 00 01 02 03 04 05       00
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    11 12 13 14 15       10 11
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       22 23 24 25       20 21 22
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          33 34 35       30 31 32 33
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             44 45       40 41 42 43 44
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                55       50 51 52 53 54 55
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Let TRANSR = \(aqN\(aq. RFP holds AP as follows:
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For UPLO = \(aqU\(aq the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
conjugate-transpose of the first three columns of AP upper.
For UPLO = \(aqL\(aq the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
conjugate-transpose of the last three columns of AP lower.
To denote conjugate we place -- above the element. This covers the
case N even and TRANSR = \(aqN\(aq.
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       RFP A                   RFP A
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                              -- -- --
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      03 04 05                33 43 53
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                                 -- --
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      13 14 15                00 44 54
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                                    --
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      23 24 25                10 11 55
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      33 34 35                20 21 22
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      --
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      00 44 45                30 31 32
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      -- --
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      01 11 55                40 41 42
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      -- -- --
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      02 12 22                50 51 52
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Now let TRANSR = \(aqC\(aq. RFP A in both UPLO cases is just the conjugate-
transpose of RFP A above. One therefore gets:
.br
         RFP A                   RFP A
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   -- -- -- --                -- -- -- -- -- --
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   03 13 23 33 00 01 02    33 00 10 20 30 40 50
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   -- -- -- -- --                -- -- -- -- --
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   04 14 24 34 44 11 12    43 44 11 21 31 41 51
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   -- -- -- -- -- --                -- -- -- --
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   05 15 25 35 45 55 22    53 54 55 22 32 42 52
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We next  consider Standard Packed Format when N is odd.
.br
We give an example where N = 5.
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   AP is Upper                 AP is Lower
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 00 01 02 03 04              00
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    11 12 13 14              10 11
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       22 23 24              20 21 22
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          33 34              30 31 32 33
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             44              40 41 42 43 44
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Let TRANSR = \(aqN\(aq. RFP holds AP as follows:
.br
For UPLO = \(aqU\(aq the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
conjugate-transpose of the first two   columns of AP upper.
For UPLO = \(aqL\(aq the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
conjugate-transpose of the last two   columns of AP lower.
To denote conjugate we place -- above the element. This covers the
case N odd  and TRANSR = \(aqN\(aq.
.br
       RFP A                   RFP A
.br
                                 -- --
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      02 03 04                00 33 43
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                                    --
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      12 13 14                10 11 44
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      22 23 24                20 21 22
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      --
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      00 33 34                30 31 32
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      -- --
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      01 11 44                40 41 42
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Now let TRANSR = \(aqC\(aq. RFP A in both UPLO cases is just the conjugate-
transpose of RFP A above. One therefore gets:
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         RFP A                   RFP A
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   -- -- --                   -- -- -- -- -- --
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   02 12 22 00 01             00 10 20 30 40 50
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   -- -- -- --                   -- -- -- -- --
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   03 13 23 33 11             33 11 21 31 41 51
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   -- -- -- -- --                   -- -- -- --
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   04 14 24 34 44             43 44 22 32 42 52
.br
